Study on the Lateral Dynamic Impedance of Pile Groups in Transversely Isotropic Soil Using Novak ’ s Plane Model

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Introduction
The research on the dynamic characteristics of pile foundation is related to the safety and stability of building structures.Therefore, since the 1960s, Novak [1], Novak and Nogami [2], Nogami and Novák [3], El Naggar and Novak [4], Wu et al. [5], Zheng et al. [6], Cui et al. [7], Zhang et al. [8], and others have conducted systematic theoretical research on the vibration of single pile.In practical engineering, pile foundation usually appears in the form of pile groups.Under the dynamic excitation of wind, waves, and earthquake, the dynamic response of pile groups is very complex, so the research on the dynamic characteristics of pile groups is relatively late.The research on the dynamic response of pile groups under various dynamic loads has important theoretical value and engineering significance for ensuring the stability and safety of the structure.
At present, the main research methods on the dynamic characteristics of pile groups are direct method and superposition method based on interaction factors.The direct method mainly adopts continuum mechanics method, finite element method, or boundary element method.Tajimi [9] and Koo et al. [10] studied the vibration of pile groups and pile group-structure by using the continuum mechanics method and the wave propagation theory.Sen et al. [11] proposed a boundary element formula for dynamic analysis of single pile and pile groups under axial and lateral loads.Maeso et al. [12] proposed a three-dimensional boundary element model for calculating the dynamic stiffness coefficient of pile groups in two-phase saturated elastic soil.Zhou and Wang [13] used the three-dimensional wave principle of saturated soil proposed by Biot and the Muki and Sternberg's methods to study the dynamic response of pile groups in saturated soil under horizontal harmonic loads.Fattah et al. [14] studied the dynamic behavior of pile group model in two-layer sandy soil subjected to lateral earthquake excitation.Wang and Ai [15] studied the vertical vibration of pile groups with rigid caps in multilayer porous-saturated soil based on the finite element method, and the pile is regarded as a one-dimensional rod.Ma et al. [16] considered the soil as a three-dimensional axisymmetric continuum, considered the effects of radial and vertical displacement and saturation on the dynamic shear modulus of soil, and studied the vertical dynamic impedance of end-bearing pile groups in homogeneous unsaturated soil.Finite element method and boundary element method need special analysis programs, and the amount of calculation is huge, so they are difficult to be applied to large-scale problems.At present, the interaction factor method is the most effective method to calculate the dynamic impedance of pile groups.Poulos [17] first put forward the concept of interaction factor and carried out the static calculation of pile groups on this basis.Kaynia [18] extended the principle of static interaction factor proposed by Poulos to the problem of dynamic interaction of pile groups and put forward the concept of dynamic interaction for the first time.Its solution is generally considered as an exact solution.Based on the simplified plane strain model, Gazetas and Dobry [19] gave an approximate expression of the wave displacement attenuation function of soil layer.Gazetas and Makris [20] and Makris and Gazetas [21] used the dynamic Winkler model to put forward a simple analysis method for calculating the axial and lateral dynamic steadystate responses of frictional pile groups in homogeneous foundation.This method further remedied the defect of Gazetas and Dobry in the calculation of dynamic interaction factors and gave the three-step method of pile-soil-pile interaction to specifically calculate and analyze the dynamic characteristics of pile groups.Liu et al. [22] used a simple Winkler model to derive the pile-soil-pile dynamic interaction factor and studied the dynamic response of partially buried pile groups in layered saturated soil under horizontal simple harmonic loads.Zhang et al. [23] adopted the three-phase porous elastic medium motion theory of unsaturated soil and the superposition method based on dynamic interaction factors to give the dynamic impedance of pile groups in unsaturated soil under horizontal vibration load in the frequency domain.In most of the previous studies on the dynamic properties of single pile and pile groups, the soil around the pile is regarded as an isotropic medium.
In the process of soil deposition, due to the orientation relationship of flat particles and the directionality of their arrangement, the properties (elastic modulus, shear modulus, and Poisson's ratio) of soil in the vertical and lateral directions are different, and the soil shows different characteristics in all directions.The vertical modulus of soil is often smaller than the lateral modulus, but the lateral modulus is isotropic.Therefore, regarding this kind of soil as transversely isotropic soil is more in line with the engineering practice.Chen et al. [24], Shahmohamadi et al. [25], Zheng et al. [6], Li and Ai [26], and Ye and Yong Ai [27] studied the torsional vibration of a single pile in transversely isotropic soil.Due to the complexity of lateral and vertical vibration of transversely isotropic soil and the difficulty of mathematical solution, there are few studies on lateral and vertical vibration of transversely isotropic soil and less on pile-pile dynamic interaction and vibration of pile groups in transversely isotropic soil.Cui et al. [28] provided the analytical solution for horizontal vibration of end-bearing single pile in radially heterogeneous saturated soil based on Biot's dynamic equations and Novak's thin-layer theory.The existing attenuation function of displacement of homogeneous soil and the pile-pile dynamic interaction factor are not suitable for the study of pile-pile dynamic interaction and dynamic characteristics of pile groups in transversely isotropic soil.In this paper, based on the wave propagation theory and Novak's plane assumption, the attenuation function of displacement of homogeneous soil and the pile-pile dynamic interaction factor are extended to the transversely isotropic soil, the lateral vibration of pile groups in transversely isotropic soil is studied, and the lateral dynamic impedance of pile groups is given.

Basic Equations
According to the elastic theory, the lateral dynamic control equation of the soil around the pile is (Luan et al. [29]) In order to consider the anisotropy of the soil around the pile, the three-dimensional constitutive relationship of transversely isotropic elastic medium proposed by Ding et al. [30] is used to describe the stress-strain relationship of the soil, i.e., According to the plane assumption proposed by Novak and Sachs [31], the soil layer is regarded as composed of many infinite thin soil layers with a circular hole, and the diameter of the hole is d.These thin layers are independent of each other, and it is considered that the soil layer displacement is independent of the coordinate z, that is, ∂u r /∂z = 0 and ∂u θ /∂z = 0.The pile-soil vibration is small deformation and assumed complete contact between pile and soil and without relative sliding and separation.Because the pilesoil system is small deformation, this assumption is rationality.Although Novak plane model ignores the interaction between soil layers, it has been widely used in engineering because of its simplicity, practicality, and high accuracy.If the vertical displacement of transversely isotropic soil is neglected, the corresponding strain displacement relationship is Here, u r and u θ are the radial and circumferential displacements of transversely isotropic soil.From formulas (1)-(3), the lateral dynamic equation of transversely isotropic soil expressed by radial and circumferential displacements is 3. Lateral Displacement Attenuation Function of Transversely Isotropic Free Field

Solution of Attenuation Function of Lateral
Displacement.When the transversely isotropic soil-pile sys-tem vibrates in a steady state under the simple harmonic load at pile top, the radial displacement and circumferential displacement of the soil meet u r = ũr e iωt , u θ = ũθ e iωt , ũr and ũθ are the amplitudes of the radial and circumferential displacement of the transversely isotropic soil, respectively; i is imaginary unit, and ω is load frequency.Following formulas can be obtained by substituting them into Equations ( 4) and ( 5) and performing dimensionless operation: where represent the vertical shear wave velocity of soil mass.The following potential function is introduced to decouple Equations ( 6) and (7): We obtained In which Considering that the soil displacement at infinity is zero, and the parity of radial displacement and circumferential displacement, it is assumed that the pile bottom is bedrock, and the pile and bedrock are completely fixed.Solving Equations ( 9) and (10) with the method of separating variables will have where q 2 = ω 2 /c 11 G hv , and 1 ð⋅Þ is the Hankel function of the second kind of order 1, and A and B are undetermined coefficients.Substituting Equations ( 12) and ( 13) into Equation ( 8), it can be obtained that the radial displacement and circumferential displacement of transversely isotropic soil layer are u r = qH In which, H ð2Þ 0 ð⋅Þ is the Hankel function of the second kind of order 0. It is assumed that the dimensionless lateral displacement of pile foundation at the pile-soil contact surface is u p ; then, there are boundary conditions: From Equations ( 14)-( 16), the undetermined coefficient A and B can be determined, and then, the radial displacement and circumferential displacement of the transversely isotropic soil layer can be determined as In which Under the action of simple harmonic load on the top of the active pile, the vibration of the active pile will cause the vibration of the soil around the pile and then produce the outward spreading cylindrical wave.It is assumed that the radiation wave follows the plane strain assumption, that is, the radiation wave only propagates along the lateral direction in each soil layer, but the attenuation function delays with the propagation distance S and the load direction angle θ in the propagation process.The displacement attenuation function of soil layer at θ = 0 0 and θ = 90 0 can be defined by Equations ( 17) and (18), that is In which, S = S/d.Considering the lateral displacement, radial displacement, and circumferential displacement, the attenuation function of lateral displacement amplitude of free field can be obtained as In practical engineering, sometimes, for the convenience of research, the simplified formula given by Gazetas and Dobry [19] is used: where v La = 3:4v s /πð1 − vÞ.ξ is the viscosity coefficient of soil mass, and v s is the shear wave velocity of soil mass.The simplified formula given by Dobry and Gazetas is applicable to homogeneous soil, and the plane strain model solution is required for transversely isotropic soil.

Pile-Pile Lateral Dynamic Interaction Factor in Transversely Isotropic Soil
In order to use the pile-pie superposition principle to solve the dynamic impedance of pile groups, it is necessary to cal-culate the pile-pile lateral dynamic interaction factor in transversely isotropic soil.It is assumed that the geometric dimensions and material properties of the active pile and the passive pile are the same (as shown in Figure 5), both of which are circular sections; the diameter is d, and the length is H. Since the size of the pile is usually small relative to the wavelength, when the wave generated by the active pile propagates to the passive pile, the displacement caused by each point in the section direction of the passive pile is

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Journal of Applied Mathematics almost the same, so the radial size of the passive pile can be ignored and replaced by its axis.From Equations ( 1) and ( 2), it can be obtained that the dimensionless quantities of radial stress and shear stress of transversely isotropic soil are where σ rr = σrr /G v , σ rθ = σrθ /G v , σrr , and σrθ are the amplitudes of radial and shear stresses σ rr and σ rθ , respectively.From Equations ( 17), ( 18), (24), and ( 25), the radial stress and shear stress of transversely isotropic soil are obtained, respectively.
The dynamic interaction between pile and soil is simulated by Winkler spring-damper model.In order to obtain the stiffness coefficient and damping coefficient of Winkler model, the unit thickness soil layer is taken as the research object, and the force of the transversely isotropic soil around the pile on the pile along the lateral x direction is In which, When generating unit lateral displacement, the required lateral force πT is the lateral impedance of the soil layer, that is, the stiffness and damping coefficient of the spring meet Firstly, taking the active pile as the research object and considering the effect of soil around the pile, a dimensionless lateral dynamic equation of active pile in transversely isotropic soil is established as where λ 4 = 16T − 4 ρ p ω 2 / E p , ρ p = ρ p /ρ, E p = E p /G v , U 11 ð zÞ = ũ11 ðzÞ/d, and ũ11 ðzÞ are the amplitude of the lateral displacement of the active pile.From the initial parameter method (Yan and Liu [32]), the general solution of Equation ( 31) is In which, , φ 11 ð0Þ, Q 11 ð0Þ, andM 11 ð0Þ are the displacement and internal force at the top of the active pile.F 1 , F 2 , F 3 , and F 4 are Krylov functions, and their derivative relations satisfy Taking the passive pile as the research object and considering the influence of the lateral displacement of free soil at the passive pile caused by the active pile, the lateral dynamic equation of the passive pile can be established as Similarly, the general solution of Equation ( 33) can be obtained from the initial parameter method as From the lateral displacement of active pile and passive pile, the rotation angle, shear force, and bending moment of active pile and passive pile can be obtained, and then, the relationship of the displacement, rotation angle, shear force, and bending moment of active pile and passive pile between pile top and pile bottom can be obtained as In which, δ = H/d is the pile length diameter ratio, and For the convenience of derivation, the matrix ½T 1 , ½T 2 , and ½T 3 are divided into four parts, i.e., Assuming that both the active pile and the passive pile have fixed ends, and considering the boundary conditions of the active pile, the stiffness matrix and flexibility matrix of a single pile can be obtained as The element K s ð1, 1Þ of the stiffness matrix ½K s is the lateral dynamic impedance of an active single pile.Considering the boundary conditions of passive piles, the pile-pile relationship matrix under lateral vibration can be obtained from Equation (36), that is According to the definition of interaction factor, the lateral dynamic interaction factor of pile-pile in transversely isotropic soil is where αð1, 1Þ and αð1, 2Þ are the element of pile-pile interaction matrix during lateral vibration.

Lateral Dynamic Impedance of Pile
Groups in Transversely Isotropic Soil 5.1.Solution of Lateral Vibration of Pile Groups.The lateral vibration of pile groups in transversely isotropic soil under rigid caps is investigated.The number of piles is n, the physical and geometric properties of each pile are the same, and the pile foundation and bearing platform are symmetrically distributed.Regardless of the mass of rigid cap, there is lateral harmonic load acting on the cap, and the dimensionless load amplitude is P. Since the pile top is a rigid cap, the lateral displacement at the pile groups top is equal to the lateral displacements of each pile at top, we have where U G and U i are the dimensionless quantities of the lateral displacement amplitude of the pile groups and the i-th pile top, and P j is the lateral load shared by the j-th pile, α uP,ij are lateral dynamic interaction factor between the i -th pile and the j-th pile.Taking the rigid cap as the research object, the following equilibrium equation can be obtained: By solving Equations ( 45) and ( 46), the lateral displacement U G and U i of the pile groups and the i-th pile at top and the lateral load P j of each pile at top can be obtained, and then, the lateral dynamic impedance of the pile groups in transversely isotropic soil can be obtained.
where K h and C h are the lateral dynamic stiffness and dynamic damping of pile groups in transversely isotropic soil.

Examples of 3 × 3 Pile Groups in Transversely Isotropic
Soil.Figures 6-9 show the variation curves of lateral dynamic stiffness and dynamic damping of pile groups in transversely isotropic soil with frequency, and the parameter value is G hv = 1:5, μ hh = 0:4, μ vh = 0:35, ρ p /ρ = 2:0, E p /G v = 1000, and H/d = 20.Figure 6 gives the comparison curves of lateral dynamic stiffness and dynamic damping of pile groups in homogeneous soil (G hv = 1:0, μ hh = μ vh = 0:35) and transverse isotropic soil.Obviously, the lateral dynamic impedance of pile groups in transversely isotropic soil can degenerate to that in homogeneous soil, which shows the correctness of the analysis method in this paper.Poisson's ratio μ hh of orthogonal lateral strain caused by lateral stress and Poisson's ratio μ vh of lateral strain caused by vertical stress 9 Journal of Applied Mathematics have little effect on the lateral dynamic impedance of pile groups, only on the value of the curve of lateral dynamic stiffness and dynamic damping varying with frequency.
The ratio G hv of the shear modulus in the lateral plane to the shear modulus in the vertical plane has a great impact on the lateral dynamic stiffness and dynamic damping of pile groups (Figures 6 and 7).At high frequencies, the greater the shear modulus ratio G hv in both directions, the greater the lateral dynamic stiffness and dynamic damping of pile groups, while at low frequencies, it is the opposite.It can be seen that the difference of mechanical properties on different surfaces of soil around the pile has a great influence on the lateral vibration of pile groups in transversely isotropic soil, and the influence of anisotropy of soil around the pile should not be ignored when studying the lateral vibration of pile groups.
The pile spacing S/d has great influence on the lateral vibration of pile groups in transversely isotropic soil, as  ), the change of lateral dynamic impedance of pile groups in transversely isotropic soil with frequency is relatively small, and the dynamic stiffness gradually decreases with the increase of frequency and tends to negative value, while the change of dynamic damping with frequency is small.At this time, the lateral vibration of pile groups is similar to that of embedded foundation.The curves of lateral dynamic stiffness and dynamic damping of pile groups in transversely isotropic soil varying with frequency will appear peaks and valleys when the pile spacing increases.The larger the pile spacing is, the more severe and complex the variation curves of lateral dynamic stiffness and dynamic damping with frequency are.The influence of the ratio E p /G v of the elastic modulus of the pile to the vertical plane shear modulus of the soil around the pile on the lateral vibration of pile groups in transversely isotropic soil is shown in Figure 9.At low frequency, the influence of modulus ratio on lateral dynamic   11 Journal of Applied Mathematics stiffness is relatively small, while at high frequency, the influence is larger, while the influence of modulus ratio on dynamic damping is larger and is not affected by frequency.The greater the elastic modulus of the pile, the greater the stiffness of the pile, the smaller the lateral displacement of the pile, and the greater the lateral dynamic stiffness and dynamic damping at the pile top.

Conclusions
Based on Novak's plane assumption and considering the anisotropy of the soil around the pile, the concept of pilepile dynamic interaction factor is extended to the transversely isotropic soil, and the lateral vibration of pile groups is studied.The following conclusions are obtained through numerical analysis: (1) the anisotropy of the soil around the pile has a great influence on the propagation of the radiation wave, and the influence of the anisotropy cannot be ignored.The attenuation function of the lateral displacement obtained by the simplified solution of Dobry and Gazetas is different from the solution of the plane model; (2) when the pile spacing is small, the real and imaginary parts of the lateral displacement attenuation function of the soil change slowly and stably with frequency, and when the pile spacing is large, the curve fluctuates greatly with frequency; (3) Poisson's ratio μ hh of orthogonal lateral strain caused by lateral stress and Poisson's ratio μ vh of lateral strain caused by vertical stress only have certain influence on the peak value of the curve of lateral dynamic stiffness and dynamic damping varying with frequency; (4) the ratio of G hv the shear modulus in the lateral plane to the shear modulus in the vertical plane and the pile spacing S/d have a great impact on the lateral vibration of pile groups in transversely isotropic soil.The impact of the difference of mechanical properties of the soil around the pile on the lateral vibration of pile groups in transversely isotropic soil should not be ignored.(5) The ratio E p /G v of elastic modulus of pile to vertical plane shear modulus of soil has an effect on the lateral dynamic stiffness of pile groups in transversely isotropic soil, and the influence is related to frequency, but the influence of E p /G v on dynamic damping is not affected by frequency.(6) In this paper, Novak's plane model is used to study the lateral vibration of pile groups in transversely isotropic soil.Some simplification and assumptions are made to the problem, and the vertical displacement of transversely isotropic soil is ignored.Although it can meet the needs of engineering, its accuracy is a little lower than that of three-dimensional model.Three-dimensional model can be used to study the lateral vibration of pile groups in transversely isotropic soil by finite element method or boundary element method and to carry out comparative study with engineering examples.

GGFigure 6 :
Figure 6: (a) Curves of lateral dynamic stiffness of pile groups in transversely isotropic soil varying with frequency.(b) Curves of lateral dynamic damping of pile groups in transversely isotropic soil varying with frequency.

Figure 7 :
Figure 7: (a) Influence of shear modulus ratio G hv on lateral dynamic stiffness of pile groups in transversely isotropic soil.(b) Influence of shear modulus ratio G hv on lateral dynamic damping of pile groups in transversely isotropic soil.

Figure 8 :
Figure 8: (a) Influence of pile spacing S/d on lateral dynamic stiffness of pile groups in transversely isotropic soil.(b) Influence of pile spacing S/d on lateral dynamic damping of pile groups in transversely isotropic soil.

Figure 9 :
Figure 9: (a) Influence of pile-soil modulus ratio E v /G v on lateral dynamic stiffness of pile groups in transversely isotropic soil.(b) Influence of pile-soil modulus ratio E v /G v on lateral dynamic damping of pile groups in transversely isotropic soil.
where σ rr , σ θθ , σ zz , σ rθ , σ rz , σ θz are the radial stress, circumferential stress, vertical stress, and shear stress of transversely isotropic soil; ε rr , ε θθ , ε zz , ε rθ , ε rz , ε θz are the radial strain, circumferential strain, vertical strain, and shear strain of transversely isotropic soil mass.C 11 , C 22 , C 33 , C 12 , C 13 , C 23 , C 44 , C 55 , and C 66 are the elastic constants of transversely isotropic soil mass, which meet Journal of Applied MathematicsE h ðμ hh + μ hv μ vh Þ/ð1 + μ hh Þð1 − μ hh − 2μ hv μ vh Þ, and C 66 = 1/2 ðC 11 − C 12 Þ = G h .E hand E v are the lateral and vertical elastic modulus, respectively, G h and G v are the shear modulus in the lateral and vertical planes, respectively, μ hh is Poisson's ratio of orthogonal lateral strain caused by lateral stress, μ hv is the Poisson's ratio of vertical strain caused by lateral stress, μ vh is the Poisson's ratio of lateral strain caused by vertical stress, and